# Evaluate the integral. \int_{0}^{1}\int_{0}^{\pi}\int_{0}^{\pi}y*\cos(z)dx dy dz

Evaluate the integral.
${\int }_{0}^{1}{\int }_{0}^{\pi }{\int }_{0}^{\pi }y\cdot \mathrm{cos}\left(z\right)dxdydz$
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Demi-Leigh Barrera
Step 1
Given that: ${\int }_{0}^{1}{\int }_{0}^{\pi }{\int }_{0}^{\pi }y\cdot \mathrm{cos}\left(z\right)dxdydz$
Step 2
As the given integral have constant limits.
So, each integral can be evaluated separately.
Thus, ${\int }_{0}^{1}{\int }_{0}^{\pi }{\int }_{0}^{\pi }y\cdot \mathrm{cos}\left(z\right)dxdydz={\int }_{0}^{1}1dx{\int }_{0}^{\pi }ydy{\int }_{0}^{\pi }\mathrm{cos}zdz$
$={\left[x\right]}_{0}^{1}\cdot {\left[\frac{{y}^{2}}{2}\right]}_{0}^{\pi }{\left[\mathrm{sin}z\right]}_{0}^{\pi }$
$=\left[1-0\right]\cdot \left[\frac{{\pi }^{2}-{0}^{2}}{2}\right]\left[\mathrm{sin}\pi -\mathrm{sin}0\right]$
$=1×\frac{{\pi }^{2}}{2}×0$
=0
So, ${\int }_{0}^{1}{\int }_{0}^{\pi }{\int }_{0}^{\pi }y\cdot \mathrm{cos}\left(z\right)dxdydz=0$